Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:
f (x) = x²
Answers
Answer:
ANSWER
Maximum or minimum can be seen by using derivatives.
Step 1: First find first derivative of the function
Step 2: Put it equal to zero and find x were first derivative is zero
Step 3: Now find second derivative
Step 4: Put x for which first derivative was zero in equation of second derivative
Step 5: If second derivative is greater than zero then function takes minimum value at that x and if second derivative is negative then function will take maximum value at that x. If Second derivative is zero them it means that this is the point of inflection.
g
′
(x)=
2
1
−
x
2
1
Putting this equal to zero, we get
g
′
(x)=
2
1
−
x
2
1
=0
⇒x=±2.
Please note that −2 is not in the domain of given function so it is of no use to us
Now let's see the double derivative of this function.
f
′′
(x)=
x
3
4
At x=2
f
′′
(2)=
2
3
4
=
2
1
Which is positive, so function will take minimum value at x=2
Minimum value is given by
f(x)=
2
2
+
2
2